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All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Automated Symbolic Upscaling: 2. Model Generation for Extended Applicability Regimes

Abstract In this second part of the two paper series, we detail an algorithmic procedure for systematically implementing the generalized closure form strategy presented in Part 1. This strategy extends the applicability of homogenized models with respect to classical homogenization theory, as demonstrated in Part 1 where upscaled models are rigorously derived in moderately reactive physical regimes. After encoding the algorithm into Symbolica, an automated upscaling framework, we upscale two reactive mass transport problems and numerically validate the resulting nonlinear homogenized models by showing the absolute error estimates predicted by homogenization theory are satisfied. In both problems, nontrivial closure forms and closure problems are automatically formulated using the encoded strategy with no human interaction, nor prior knowledge regarding the closure required for the systems. We hope these demonstrations spark further interest in automated analytical frameworks for multiscale modeling, as such capabilities are invaluable for generating rigorous multiscale models of complex phenomena in porous media.

Pietrzyk, Kyle↗
DOE OSTI2023

Automated Symbolic Upscaling: 1. Model Generation for Extended Applicability Regimes

Abstract In porous media theory, upscaling techniques are fundamental to deriving rigorous Darcy‐scale models for flow and reactive transport in subsurface systems. Due to limitations in classical techniques, a number of ad hoc approaches have been proposed to address physical regimes in which reactive time scales are similar to, or faster than, diffusive time scales. In Part 1 of this two part series, we present a strategy for expanding the applicability of classical homogenization theory by generalizing the assumed closure form. We detail the implementation of this strategy on two reactive mass transport problems with moderately reactive physics. The strategy produces nontrivial homogenized models with emergent terms and effective parameters that couple reactive, diffusive, and advective transport. The differences in equation forms between the macroscopic and pore‐scale descriptions advise caution to further studies where the forms of macroscopic equations are assumed, as opposed to rigorously derived. Numerical validation is provided for each problem to show that the error estimates of homogenization theory are satisfied, and to justify the implemented strategy. In Part 2, the presented strategy is automated using symbolic computing to expedite its implementation.

Pietrzyk, Kyle↗
DOE OSTI2017

Numerical Multilevel Upscaling for Incompressible Flow in Reservoir Simulation: An Element-Based Algebraic Multigrid (AMGe) Approach

Here, we study the application of a finite element numerical upscaling technique to the incompressible two-phase porous media total velocity formulation. Specifically, an element-agglomeration-based algebraic multigrid (AMGe) technique with improved approximation properties is used, for the first time, to generate upscaled and accurate coarse systems for the reservoir simulation equations. The upscaling technique is applied to both the mixed system for velocity and pressure and to the hyperbolic transport equations, providing fully upscaled systems. By introducing additional degrees of freedom associated with nonplanar interfaces between agglomerates, the coarse velocity space has guaranteed approximation properties. The employed AMGe technique provides coarse spaces with desirable local mass conservation and stability properties analogous to the original pair of Raviart--Thomas and piecewise discontinuous polynomial spaces, resulting in strong mass conservation for the upscaled systems. Due to the guaranteed approximation properties and the generic nature of the AMGe method, recursive multilevel upscaling is automatically obtained. Furthermore, this technique works for both structured and unstructured meshes. Multiscale mixed finite elements exhibit accuracy for general unstructured meshes but do not in general lead to nested hierarchy of spaces. Multiscale multilevel mimetic finite differences generate nested spaces but lack the adaptivity of the flux representation on coarser levels that the proposed AMGe approach offers. Thus, the proposed approach can be seen as a rigorous bridge that merges the best properties of these two existing methods. The accuracy and stability of the studied multilevel AMGe upscaling technique is demonstrated on two challenging test cases.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

The Method of Finite Averages: A rigorous upscaling methodology for heterogeneous porous media

Rigorous upscaling techniques offer accurate and computationally-efficient strategies for modeling the average behaviors of multi-physical, multiscale phenomena in geological porous media. However, such techniques often rely on a variety of methodological assumptions that prohibit their rigorous application to practical systems (e.g., systems involving heterogeneous porous media, system-scale boundary conditions, and fine-scale dynamics that are not diffusion-dominant). In this work, we aim to formulate an upscaling methodology with few methodological assumptions to provide high levels of model generality and foster the utilization of rigorously-derived upscaled models in practice. In particular, we introduce the Method of Finite Averages (MoFA), a novel upscaling methodology for rigorously modeling heterogeneous porous media and system-scale boundary conditions. We then detail MoFA’s implementation for the advective–diffusive transport of a single species and compare the methodology with classic numerical techniques, as well as other rigorous upscaling techniques, to highlight MoFA’s unique combination of rigor and generality. We then validate the derived model while demonstrating its benefits in three numerical experiments. The results suggest that (1.) the applicability and a priori error guarantees of MoFA models do not directly depend on system geometry, (2.) a model’s applicability and error guarantees can be can arbitrarily expanded and reduced, respectively, with further computational expense, and (3.) downscaling with MoFA provides an efficient strategy for generating accurate pore-scale solutions from upscaled results. Ultimately, the results evidence that upscaled models can be rigorously derived for heterogeneous porous media systems and resolved in a fraction of the time it takes to perform the equivalent pore-scale simulations.

58 GEOSCIENCES↗
DOE OSTI2021

Developing Upscaling Approach for Swarming Hydraulic Fractures Observed at Hydraulic Fracturing Test Site through Multiscale Simulations

We show that this work aims to address a challenge posed by recent observations of tightly spaced hydraulic fractures in core samples from the hydraulic fracturing test site (HFTS) in the Middle Wolfcamp Formation. Many fractures in retrieved cores have subfoot spacing, which is at odds with conventional models in which usually one hydraulic fracture is initiated per cluster. Models assuming a single fracture at each cluster, although a common practice, often predict excessive fracture propagation that is inconsistent with microseismic observation. Here, we aim to develop a numerical approach to effectively account for densely spaced hydraulic fractures in field-scale simulations. Because it is impractical to explicitly model all aforementioned fractures, we develop a new upscaling law that enables existing simulation tools to predict reservoir response to fracture swarms. The upscaling law is derived based on an energy equivalence argument and validated through multiscale simulations using a high-fidelity code, GEOS. The swarming fractures are first modeled with a spacing that is much smaller than the cluster spacing; these fractures are then approximated by an upscaled, single fracture based on the proposed upscaling law. The upscaled fracture is shown to successfully match the energy input rate and produce the total fracture aperture and average propagation length of the explicitly simulated swarm. Afterward, the upscaling approach is further implemented in 3D field-scale simulations and validated against the HFTS microseismic data of a horizontal well. Our results show that hydraulic fracture swarming can significantly affect fracture propagation behaviors compared with the propagation of single fractures as assumed by conventional modeling approaches. Under the considered situations, the conventional treatment yields fast propagation speed that far exceeds that indicated by the microseismic data. We also illustrate that this discrepancy can be reduced readily through the implementation of the upscaling law. Our results demonstrate the importance of accounting for the fracture swarming effect in field-scale simulations and the efficacy of this approach to enable realistic predictions of reservoir responses to fracture swarms, without the need to model tightly spaced fractures individually.

02 PETROLEUM↗
DOE OSTI2023

Can upscaling ground nadir SIF to eddy covariance footprint improve the relationship between SIF and GPP in croplands?

Ground solar-induced chlorophyll fluorescence (SIF) is important for the mechanistic understanding of the dynamics of vegetation gross primary production (GPP) at fine spatiotemporal scales. However, eddy covariance (EC) observations generally cover larger footprint areas than ground SIF observations (a bare fiber with nadir), and this footprint mismatch between nadir SIF and GPP could complicate the canopy SIF-GPP relationships. Here, we upscaled nadir SIF observations to EC footprint and investigated the change in SIF-GPP relationships after the upscaling in cropland. We included 13 site-years data in our study, with seven site-years corn, four site-years soybeans, and two site-years miscanthus, all located in the US Corn Belt. All sites’ crop nadir SIF observations collected from the automated FluoSpec2 system (a hemispheric-nadir system) were upscaled to the GPP footprint-based SIF using vegetation indices (VIs) calculated from high spatiotemporal satellite reflectance data. We found that SIF-GPP relationships were not substantially changed after upscaling nadir SIF to GPP footprint at our crop sites planted with corn, soybean, and miscanthus, with R 2 change after the upscaling ranging from -0.007 to 0.051 and root mean square error (RMSE) difference from -0.658 to 0.095 umol m –2 s –1 relative to original nadir SIF-GPP relationships across all the site-years. The variation of the SIF-GPP relationship within each species across different site-years was similar between the original nadir SIF and upscaled SIF. Different VIs, EC footprint models, and satellite data led to marginal differences in the SIF-GPP relationships when upscaling nadir SIF to EC footprint. Furthermore, our study provided a methodological framework to correct this spatial mismatch between ground nadir SIF and GPP observations for croplands and potentially for other ecosystems. Our results also demonstrated that the spatial mismatch between ground nadir SIF and GPP might not significantly affect the SIF-GPP relationship in cropland that are largely homogeneous.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2020

Adaptive Multirate Mass Transfer (aMMT) Model: A New Approach to Upscale Regional‐Scale Transport Under Transient Flow Conditions

Abstract The long‐term evaluation of regional‐scale groundwater quality needs efficient upscaling methods for transient flow. Upscaling techniques, such as the Multirate Mass Transfer (MRMT) method with constant upscaling parameters, have been used for transport with steady‐state flow, yet the upscaling parameters (i.e., rate coefficients) may be time dependent. This study proposed and validated an adaptive MRMT (aMMT) method by allowing the mass transfer coefficients in MRMT to change with the flow field. Advective‐dispersive contaminant transport simulated in a 3‐D heterogeneous medium was used as a reference solution. Equivalent transport under homogeneous flow conditions was evaluated by applying the MRMT and aMMT models for upscaling. The relationship between mass transfer coefficients and flow rates was fitted under steady‐state flow driven by various hydraulic gradients. A power law relationship was obtained, which was then used to update the mass transfer coefficients in each stress period under transient flow conditions in the aMMT method. Results indicated that for advection‐dominated transport, both the MRMT and aMMT methods can upscale the anomalous transport dynamics affected by subgrid heterogeneity under transient flow conditions. Whereas for diffusion‐dominated systems, the MRMT model failed to capture the tails of tracer breakthrough curves after the boundary condition changed, but the results from the aMMT model were significantly improved. However, if the overall flow direction changed, both MRMT and aMMT failed to represent the breakthrough curve tail generated by the heterogeneous system. The results point toward a promising path for upscaling transport in complex aquifers with transient flow.

Guo, Zhilin↗
DOE OSTI2017

Upscaling instantaneous to daily evapotranspiration using modelled daily shortwave radiation for remote sensing applications: An artificial neural network approach

Upscaling instantaneous evapotranspiration retrieved at any specific time-of-day (ET i ) to daily evapotranspiration (ET d ) is a key challenge in mapping regional ET using polar orbiting sensors. Various studies have unanimously cited the shortwave incoming radiation ( R S ) to be the most robust reference variable explaining the ratio between ET d and ET i . This study aims to contribute in ET i upscaling for global studies using the ratio between daily and instantaneous incoming shortwave radiation ( R Sd / R Si ) as a factor for converting ET i to ET d . This paper proposes an artificial neural network (ANN) machine-learning algorithm first to predict R Sd from R Si followed by using the R Sd / R Si ratio to convert ET i to ET d across different terrestrial ecosystems. Using R Si and R Sd observations from multiple sub-networks of the FLUXNET database spread across different climates and biomes (to represent inputs that would typically be obtainable from remote sensors during the overpass time) in conjunction with some astronomical variables (e.g. solar zenith angle, day length, exoatmospheric shortwave radiation), we developed the ANN model for reproducing R Sd and further used it to upscale ET i to ET d . The efficiency of the ANN is evaluated for different morning and afternoon times of day, under varying sky conditions, and also at different geographic locations. R S -based upscaled ET d produced a significant linear relation ( R 2 = 0.65 to 0.69), low bias (-0.31 to -0.56 MJ m -2 d -1 ; approx. 4 %), and good agreement (RMSE 1.55 to 1.86 MJ m -2 d -1 ; approx. 10 %) with the observed ET d , although a systematic overestimation of ET d was also noted under persistent cloudy sky conditions. Inclusion of soil moisture and rainfall information in ANN training reduced the systematic overestimation tendency in predominantly overcast days. An intercomparison with existing upscaling method at daily, 8-day, monthly, and yearly temporal resolution revealed a robust performance of the ANN-driven R S -based ET i upscaling method and was found to produce lowest RMSE under cloudy conditions. Sensitivity analysis revealed variable sensitivity of the method to biome selection and high ET d prediction errors in forest ecosystems are primarily associated with greater rainfall and cloudiness. As a result, the overall methodology appears to be promising and has substantial potential for upscaling ET i to ET d for field and regional-scale evapotranspiration mapping studies using polar orbiting satellites.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2016

Upscaling of Mixed Finite Element Discretization Problems by the Spectral AMGe Method

Here, we propose two multilevel spectral techniques for constructing coarse discretization spaces for saddle-point problems corresponding to PDEs involving a divergence constraint, with a focus on mixed finite element discretizations of scalar self-adjoint second order elliptic equations on general unstructured grids. We use element agglomeration algebraic multigrid (AMGe), which employs coarse elements that can have nonstandard shape since they are agglomerates of fine-grid elements. The coarse basis associated with each agglomerated coarse element is constructed by solving local eigenvalue problems and local mixed finite element problems. This construction leads to stable upscaled coarse spaces and guarantees the inf-sup compatibility of the upscaled discretization. Also, the approximation properties of these upscaled spaces improve by adding more local eigenfunctions to the coarse spaces. The higher accuracy comes at the cost of additional computational effort, as the sparsity of the resulting upscaled coarse discretization (referred to as operator complexity) deteriorates when we introduce additional functions in the coarse space. We also provide an efficient solver for the coarse (upscaled) saddle-point system by employing hybridization, which leads to a symmetric positive definite (s.p.d.) reduced system for the Lagrange multipliers, and to solve the latter s.p.d. system, we use our previously developed spectral AMGe solver. Numerical experiments, in both two and three dimensions, are provided to illustrate the efficiency of the proposed upscaling technique.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Upscaling permeability for three-dimensional fractured porous rocks with the multiple boundary method

Upscaling permeability of grid blocks is crucial for groundwater models. A novel upscaling method for three-dimensional fractured porous rocks is presented. The objective of the study was to compare this method with the commonly used Oda upscaling method and the volume averaging method. First, the multiple boundary method and its computational framework were defined for three-dimensional stochastic fracture networks. Then, the different upscaling methods were compared for a set of rotated fractures, for tortuous fractures, and for two discrete fracture networks. The results computed by the multiple boundary method are comparable with those of the other two methods and fit best the analytical solution for a set of rotated fractures. The errors in flow rate of the equivalent fracture model decrease when using the multiple boundary method. Furthermore, the errors of the equivalent fracture models increase from well-connected fracture networks to poorly connected ones. Finally, the diagonal components of the equivalent permeability tensors tend to follow a normal or log-normal distribution for the well-connected fracture network model with infinite fracture size. By contrast, they exhibit a power-law distribution for the poorly connected fracture network with multiple scale fractures. The study demonstrates the accuracy and the flexibility of the multiple boundary upscaling concept. This makes it attractive for being incorporated into any existing flow-based upscaling procedures, which helps in reducing the uncertainty of groundwater models.

58 GEOSCIENCES↗
DOE OSTI2020

Discrete Fracture Network Modeling to Estimate Upscaled Parameters for the Topopah Spring, Lava Flow, and Tiva Canyon Aquifers at Pahute Mesa, Nevada National Security Site

This report describes the results of Discrete Fracture Network (DFN) simulations for the Topopah Spring Aquifer (TSA), Lava Flow Aquifer, and Tiva Canyon Aquifer (TCA), at Pahute Mesa on the Nevada National Security Site (NNSS), formerly the Nevada Test Site. The research focuses on calculating upscaled groundwater flow and contaminant transport parameters using DFNs generated according to fracture characteristics observed in the TSA, LFA and TCA at Pahute Mesa. The highly fractured and heterogeneous nature of these aquifers makes them candidates for stochastic DFN modeling of radionuclide transport on a small scale with subsequent upscaling. One hundred independent DFN realizations are generated for each aquifer, and the upscaled parameters for continuum simulations of subsurface flow and transport in fractured media at Pahute Mesa are calculated. Our goal is to implement a modeling approach that can translate parameters to larger-scale models that account for local-scale flow and transport processes, such as channelization of flow and transport along a few well connected, large fractures. Additionally, to simulate advective and advective-diffusive transport through the fracture networks, the Time Domain Random Walk (TDRW) approach is applied to account for matrix diffusion into a finite half-space. Moreover, a novel approach to calculate dynamic (active) fracture surface area to reflect flow channeling is implemented. This work will improve the representation of radionuclide transport processes in largescale, regulatory-focused models by providing estimates of hard-to-measure flow and contaminant transport parameters at large scales. In this report, we (1) show recent results of flow and transport simulations on multiple DFN realizations of the TSA, LFA, TCA; (2) discuss the resulting distributions of estimated upscaled parameters; (3) describe the estimation of upscaled parameters for an equivalent parallel-plate continuum model and (4) present a comparison between simulated transport from the equivalent continuum model and an actual DFN.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

Upscaling Methods Applied to a Fine-Scale Reservoir Model

This study was conducted as part of the Southwest Regional Partnership on Carbon Sequestration (SWP) project to evaluate how upscaling fine-scale simulation models to coarse-scale simulation models impacted the results. The focus was on the Farnsworth Unit (FWU) and its Morrow' B' Sandstone reservoir, specifically the west half of the field. Due to data limitations and the geologic characteristics of the surrounding area, the upscaling was limited to the west half of the FWU rather than a broader basinscale model. The primary aim was to explore how upscaling impacts numerical simulation models, particularly regarding CO 2 -enhanced oil recovery (EOR) and storage capacity predictions. Upscaling was necessary to reduce computational demands when transitioning from high-resolution geological models to coarser grids, as large-scale simulations with finer grids can be computationally prohibitive. This study expands on previous work by the SWP to understand how additional upscaling, applied to already fine-scale numerical simulation models, affects reservoir performance simulations (Ampomah, Balch, & Grigg, 2015). This is key to understanding how loss of resolution can affect coarsescale model results that may be used for large sensitivity analyses, uncertainty quantifications, and training data for machine learning applications.

02 PETROLEUM↗
DOE OSTI2025

The rigorous upscaling of advection-dominated transport in heterogeneous porous media via the Method of Finite Averages

Systems involving advection-dominated transport through heterogeneous porous and fractured media are ubiquitous in subsurface engineering applications. However, upscaling such systems continues to challenge rigorous modeling efforts, particularly when advection is stronger than diffusion at fine spatial scales (i.e., when the Péclet number is greater than one at length scales that characterize a system’s unit-cells, representative elementary volumes, or averaging regions). Here, in this work, we propose and validate a strategy for extending the Method of Finite Averages (MoFA), a rigorous upscaling methodology for heterogeneous porous media, to upscale transport systems experiencing stronger advection than diffusion at fine scales (i.e., fine-scale Péclet numbers greater than one). We detail the strategy, the physical conditions under which it can be applied while retaining a priori modeling error guarantees, and implement the strategy to obtain a MoFA model for advective-diffusive transport that accommodates advective physics at fine spatial scales. We then perform two numerical experiments considering systems with system-scale Péclet numbers of 300 and 1000 — which correspond to fine-scale Péclet numbers of 30 and 100, respectively — to verify that the error guarantees are met under the strategy. After, we conduct a numerical study to demonstrate the strategy’s advantages over the original MoFA methodology. The results suggest that rigorously-upscaled transport models for heterogeneous porous media experiencing advective physics at finer spatial scales can be derived through MoFA and resolved orders of magnitude faster than their pore-scale counterparts. The results also suggest that the presented strategy is limited to modeling shallow concentration gradients when there are large differences between the time scales related to advection and a system’s temporally-varying boundary conditions. This limitation hinders the strategy’s practicality in modeling more advective systems, and as such, opportunity exists for developing additional strategies that accommodate rapidly-varying boundary conditions — and consequentially, steeper concentration gradients — while modeling advective systems with MoFA.

36 MATERIALS SCIENCE↗
DOE OSTI2019

Upscaling of Regional Scale Transport Under Transient Conditions: Evaluation of the Multirate Mass Transfer Model

Abstract Regional scale transport models are needed to support the long‐term evaluation of groundwater quality and to develop management strategies aiming to prevent serious groundwater degradation. The purpose of this study is to evaluate the capacity of a previously developed upscaling approach to adequately describe the main solute transport processes, including the capture of late‐time tails under changing boundary conditions. Potential factors that impact the performance of upscaling methods, including temporal variations in mass transfer rates and mass distributions, were investigated. Advective‐dispersive contaminant transport in a 3‐D heterogeneous domain was simulated and used as a reference solution. The equivalent transport under homogeneous flow conditions was then evaluated by applying the multirate mass transfer (MRMT) model. The random walk particle tracking method was used to solve the solute transport for heterogeneous and homogeneous MRMT scenarios under steady state and transient conditions. The results indicate that the MRMT model can capture the tails satisfactorily for plumes transported with ambient steady state flow fields at all studied scales using the same parameters. However, when the boundary conditions change in either local, plume, or regional scale, the mass transfer model calibrated for transport under steady state conditions cannot accurately reproduce the tailings observed for the heterogeneous scenario. The deteriorating impacts of transient boundary conditions on the upscaled model are more significant for regions where the flow fields are dramatically affected, which highlights the poor applicability of the MRMT approach for complex field settings. This finding also has implications for the suitability of other potential upscaling approaches.

Guo, Zhilin↗